The linear codes of t-designs held in the Reed-Muller and Simplex codes

08/23/2020

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by   Cunsheng Ding, et al.

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A fascinating topic of combinatorics is t-designs, which have a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs for some t ≥ 1. The purpose of this paper is to study the linear codes of some t-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of t-designs held in linear codes is presented. Open problems are also presented.